
When Euler gets it wrong
Even the greatest mathematicians sometimes make mistakes. It is only human! It even happened to one of mathematics’ legendary figures, the great Leonhard Euler, in connection with sums of powers.


Even the greatest mathematicians sometimes make mistakes. It is only human! It even happened to one of mathematics’ legendary figures, the great Leonhard Euler, in connection with sums of powers.


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We often assume that the only mistakes good mathematicians make are trivial slips caused by momentary inattention, such as errors in simple calculations. Yet some have put forward mathematical conjectures that proved false—sometimes spectacularly so.

Euler rescued from oblivion Fermat's proposition that it is impossible for the sum of two cubes to be a cube. Though the explanation he provided for it was shaky, it opened up a new framework for arithmetic.

Fermat's “little” theorem is surely one of the most fruitful results in arithmetic. Ever since it was first stated in 1640, mathematicians have made constant use of it. Euler even proposed a sweeping generalization. Let's take a closer look…

Theory tells us that a regular seventeen-sided polygon—a heptadecagon—can be constructed using only a straightedge and compass. But it gives no details of the construction, which is far from straightforward.
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